x² + y² - 2x - 4 = 0 is equivalent to (x - 1)² + y² = √5².

Line passing through the point P(4, 3) is y - 3 = m(x - 4), i.e. y = m(x - 4) + 3.

Plug in to the circle equation, and we get 0 = (1 + m²)x² + (6m - 2 - 8m²)x + (16m² - 24m + 5).

According to discriminant, b² = 4ac.

(6m - 2 - 8m²)² = 4(1 + m²)(16m² - 24m + 5), i.e. m = (9 ± √65)/4.

Plug in y = (9 ± √65)(x - 4)/4 + 3 to the circle equation to get A((11 minusorplus √65)/6, (5 ± √65)/6).

The length of AP is √13.